The word hardly has a negative meaning (almost not). Although it is not grammatically negative, it makes the statement negative in meaning.
Rule:
If the statement is negative, the question tag is positive.
If the statement is positive, the question tag is negative.
Here:
He hardly speaks to strangers = Negative meaning
Main verb = speaks (Simple Present)
Auxiliary for the tag = does
So the correct tag is:
does he?
